(a) cannot be the degree sequence of a graph:Answer: $B$ cannot be the degree sequence of any graph, because the first three vertices are connected to every other vertex in the graph – in particular, vertex 6. This means that vertex 6 should have degree at least $3$, which it doesn’t.

(b) must be the degree sequence of an eulerian graph:Answer: $D$ must be the degree sequence of a Eulerian graph because every vertex has even degree.

(c) must be the degree sequence of a hamiltonian graph:Answer: Degree sequences $C$ and $E$ must be that of a Hamiltonian graph because this graph has six vertices and every vertex has degree at least $\frac{6}{2}$. By Dirac’s theorem, it must have a Hamiltonian circuit.

(d) could be the degree sequence of a tree:Answer: $A$ could be the degree sequence of a tree because the sum of the degrees is equal to $10$, so the number of edges is $5$, which is equal to the number of vertices minus $1$.

(e) could be the degree sequence of a graph, but cannot be a planar graph:Answer: None of the graphs satisfy this condition.

7. Generating functions

Find the coefficient of $x^n$ in each of the generating functions below.